Interferometer filters with compensation structure

ABSTRACT

A Mach-Zehnder interferometer (MZI) filter comprising one or more passive compensation structures are described. The passive compensation structures yield MZI filters that are intrinsically tolerant to perturbations in waveguide dimensions and/or other ambient conditions. The use of n+1 waveguide widths can mitigate n different sources of perturbation to the filter. The use of at least three different waveguide widths for each Mach-Zehnder waveguide can alleviate sensitivity of filter performance to random width or temperature variations. A tolerance compensation portion is positioned between a first coupler section and a second coupler section, wherein the tolerance compensation portion includes a first compensation section having a second width, a second compensation section having a third width and a third compensation section having a fourth width, wherein the fourth width is greater than the third width and the third width is greater than the second width.

CROSS-REFERENCES TO OTHER APPLICATIONS

This application is a continuation of U.S. application Ser. No.16/599,526, filed Oct. 11, 2019; which is a continuation of U.S.application Ser. No. 16/514,832, filed Jul. 17, 2019, now U.S. Pat. No.10,534,130, issued Jan. 14, 2020; which claims priority to U.S.Provisional Patent Application No. 62/851,039 filed May 21, 2019, U.S.Provisional Patent Application No. 62/851,559 filed May 22, 2019 and toProvisional Patent Application No. 62/853,657 filed May 28, 2019. Thedisclosures of each are hereby incorporated by reference in entirety forall purposes.

FIELD

The described embodiments relate generally to optical filter devices.More particularly, the present embodiments relate to Mach-Zehnderinterferometer (MZI) filters that include one or more compensationstructures to compensate for variations in manufacturing tolerancesand/or temperature variations and/or other perturbations.

BACKGROUND

Currently, there are a wide variety of devices that utilize opticalcircuits for communications and/or computations. Many optical circuitsrely on one or more optical filter elements to filter out undesirableoptical frequencies, so an optical frequency range of interest can beisolated.

In some applications, an MZI filter which can include a cascaded MZIfilter, may demonstrate the theoretical capability of meeting the systemspecifications. However, when practical fabrication tolerances of theMZI filter are accounted for, the MZI filter may not be able to meet thesystem specifications without additional tuning. More specifically, anMZI filter employs two parallel waveguides and fabrication variations inthe dimensions of the waveguides can produce undesirable shifts in thefrequency response of the filter. This can lead to decreased performanceparameters of the filter and/or, the failure to meet specifications andunacceptably high yield loss.

To compensate for fabrication variations some applications employ one ormore heaters that are used to actively tune the filters using thethermo-optic effect in silicon. However, the use of heaters increasespower consumption of the circuit and may not be effective for circuitsthat operate at cryogenic temperatures. Active tuning as apost-fabrication process is another common approach to mitigatingfabrication variation, however active tuning can increase expense, maybe dependent on foundry-specific processes, and could be intractable forcircuits with numerous filters. Therefore, passive compensationstructures for MZI filters that are intrinsically tolerant toperturbations from variations in waveguide dimensions and/or otherambient conditions are desired.

SUMMARY

In some embodiments, a Mach-Zehnder interferometer (MZI) filtercomprises a first waveguide having a first length and extending from afirst coupler section to a second coupler section, the first waveguidehaving a constant first width along the first length. A second waveguidehaving a second length and extending from the first coupler section tothe second coupler section includes a tolerance compensation portionpositioned between the first coupler section and the second couplersection. The tolerance compensation portion includes a firstcompensation section having a second width, a second compensationsection having a third width and a third compensation section having afourth width, wherein the fourth width is greater than the third widthand the third width is greater than the second width. A first taperportion is positioned between the first coupler section and the firstcompensation section and transitions from the first coupler section tothe second width. A second taper portion is positioned between the firstcompensation section and the second compensation section and transitionsfrom the second width to the third width. A third taper portion ispositioned between the second compensation section and the thirdcompensation section and transitions from the third width to the fourthwidth.

In some embodiments, the first compensation section has a constantsecond width, the second compensation section has a constant third widthand the third compensation section has a constant fourth width. Invarious embodiments, the tolerance compensation portion is symmetric andincludes a fourth compensation section having the third width and afifth compensation section having the second width. In some embodiments,the tolerance compensation portion in the second waveguide is a firsttolerance compensation portion and the first waveguide includes a secondtolerance compensation portion that includes a fourth compensationsection having a fifth width, wherein the fifth width is greater thanthe first width.

In some embodiments, the first waveguide and the tolerance compensationportion form components of a tolerance compensation structure thatcompensates for a variation in a width of the first waveguide and avariation in a width of the second waveguide due to manufacturingtolerances. In various embodiments, the tolerance compensation structurereduces a shift in a frequency response of the MZI filter due to thevariation in the width of the first waveguide and the variation in thewidth of the second waveguide.

In some embodiments, a method of fabricating a Mach-Zehnderinterferometer (MZI) filter tolerant to manufacturing variationscomprises forming a substrate and forming a first waveguide on thesubstrate, the first waveguide having a first length and a firstcontinuous width along the first length, wherein the first width varieswithin a first range, and forming a second waveguide on the substrate.The second waveguide includes a manufacturing tolerance compensationportion including a first compensation section having a continuoussecond width that varies in a second range, a second compensationsection having a continuous third width that varies in a third range anda third compensation section having a continuous fourth width thatvaries in a fourth range, wherein the fourth width is greater than thethird width and the third width is greater than the second width.

In some embodiments, a first taper portion is positioned between a firstcoupler section and the first compensation section and transitions fromthe first coupler section to the second width, and a second taperportion is positioned between the first compensation section and thesecond compensation section and transitions from the second width to thethird width. A third taper portion is positioned between the secondcompensation section and the third compensation section and transitionsfrom the third width to the fourth width.

In some embodiments, the tolerance compensation portion is symmetric andincludes a fourth compensation section having the third width and afifth compensation section having the second width. In variousembodiments, the tolerance compensation portion in the second waveguideis a first tolerance compensation portion and the first waveguideincludes a second tolerance compensation portion that includes a fourthcompensation section having a fifth width, wherein the fifth width isgreater than the first width.

In some embodiments, the manufacturing tolerance compensation portionreduces a shift in a frequency response of the MZI filter caused by thesecond width varying within the second range, the third width varyingwithin the third range and the fourth width varying within the fourthrange.

In some embodiments, a Mach-Zehnder interferometer (MZI) filtercomprises a first waveguide having a first width extending between afirst coupler section and a second coupler section, and a secondwaveguide extending between the first coupler section and the secondcoupler section and including a first compensation section having asecond width, a second compensation section having a third width and athird compensation section having a fourth width, wherein the fourthwidth is greater than the third width and the third width is greaterthan the second width. In various embodiments, the MZI filter furthercomprises a first taper portion positioned between the first couplersection and the first compensation section and transitioning from thefirst coupler section to the second width. A second taper portion ispositioned between the first compensation section and the secondcompensation section and transitions from the second width to the thirdwidth. A third taper portion is positioned between the secondcompensation section and the third compensation section and transitionsfrom the third width to the fourth width.

In some embodiment, the second waveguide further includes a fourthcompensation section having the third width and a fifth compensationsection having the second width. In various embodiments, the secondwaveguide includes a fourth compensation section having the third widthand a fifth compensation section having the second width.

In some embodiments, a method for making a Mach-Zehnder interferometer(MZI) filter having a compensation section that compensates for a numberof perturbations comprises fabricating a first waveguide having a firstlength and one or more first compensation sections distributed along thefirst length, wherein each first compensation section of the one or morefirst compensation sections includes a respective width and length. Themethod further comprises fabricating a second waveguide having a secondlength and one or more second compensation sections distributed alongthe second length, wherein each second compensation section of the oneor more second compensation sections includes a respective width andlength. Wherein, a sum of the one or more first compensation sectionsand the one or more second compensation sections is greater than thenumber of perturbations.

In some embodiments, the number of perturbations is selected from amanufacturing tolerance variation in a width of each of the first andthe second waveguides, a manufacturing tolerance variation in athickness of each of the first and the second waveguides and atemperature variation in each of the first and the second waveguides.

In some embodiments, a method for making a Mach-Zehnder interferometer(MZI) filter comprises fabricating a first waveguide having a firstlength and a first continuous width, and fabricating a second waveguidehaving a second length and a plurality of widths along the secondwaveguide, wherein the first and the second waveguides simultaneouslysatisfy:

${{m\lambda_{0}} = {L_{1}\left( {{n_{1}\left( \lambda_{0} \right)} - {\Sigma_{i}{n_{i}\left( \lambda_{0} \right)}\kappa_{i}}} \right)}}{v_{FSR} = \frac{c}{L_{1}\left( {n_{\mathcal{g}1} - {\sum_{i}{n_{\mathcal{g}i}\kappa_{i}}}} \right)}}{\frac{\partial n_{1}}{\partial X_{j}} = {\Sigma_{i}\kappa_{i}\frac{\partial n_{i}}{\partial X_{j}}}}{\frac{\partial^{2}n_{1}}{{\partial X_{j}}{\partial\omega}} = {\Sigma_{i}\kappa_{i}\frac{\partial^{2}n_{i}}{{\partial X_{j}}{\partial\omega}}}}{\frac{\partial^{2}n_{1}}{\partial X_{j}^{2}} = {\Sigma_{i}\kappa_{i}\frac{\partial^{2}n_{i}}{\partial X_{j}^{2}}}}$wherein:m=an integral multiple;λ₀=wavelength of light in first and second arms;L₁=reference length of first arm;λ₀=central wavelength of light in first and second arms;L_(i)=length of i^(th) portion of second arm;κ_(i)=L_(i)/L₁;v_(FSR)=free spectral range;c=speed of light;X₁=waveguide width; andX₂=waveguide thickness.

In some embodiments, the second waveguide has a first compensationsection having a second width, a second compensation section having athird width and a third compensation section having a fourth width,wherein the fourth width is greater than the third width and the thirdwidth is greater than the second width. In various embodiments thesecond waveguide further includes a first taper portion positionedbetween a first coupler section and the first compensation section andtransitioning from the first coupler section to the second width. Asecond taper portion is positioned between the first compensationsection and the second compensation section and transitions from thesecond width to the third width. A third taper portion is positionedbetween the second compensation section and the third compensationsection and transitions from the third width to the fourth width.

To better understand the nature and advantages of the presentdisclosure, reference should be made to the following description andthe accompanying figures. It is to be understood, however, that each ofthe figures is provided for the purpose of illustration only and is notintended as a definition of the limits of the scope of the presentdisclosure. Also, as a general rule, and unless it is evident to thecontrary from the description, where elements in different figures useidentical reference numbers, the elements are generally either identicalor at least similar in function or purpose.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1A illustrates a simplified plan view of an example Mach-Zehnderinterferometer filter including a passive compensation structure,according to embodiments of the disclosure;

FIG. 1B illustrates a simplified plan view of an example Mach-Zehnderinterferometer filter including two passive compensation structures,according to embodiments of the disclosure;

FIG. 1C illustrates a simplified plan view of an example Mach-Zehnderinterferometer filter including a passive compensation structure,according to embodiments of the disclosure;

FIG. 2 illustrates a single stage of a three-waveguide cascaded thirdorder MZI based filter, according to embodiments of the disclosure;

FIG. 3 illustrates an incoherently cascaded third-order MZI filterhaving four stages, according to embodiments of the disclosure;

FIG. 4 illustrates effective index parameters as a function of waveguidewidth and height for a silicon-on-insulator waveguide, according toembodiments of the disclosure;

FIGS. 5A and 5B illustrate a plotted derivative, according toembodiments of the disclosure;

FIG. 6 illustrates standard deviations for waveguides and couplers,according to embodiments of the disclosure;

FIG. 7 illustrates the statistical behavior of four-stage cascadedthird-order MZI filter without mitigation mechanisms, according toembodiments of the disclosure;

FIG. 8 illustrates designs to minimize susceptibility to fabricationerrors, according to embodiments of the disclosure;

FIG. 9 illustrates the statistical distribution of cascaded third-orderMZI's with asymmetric arm widths, according to embodiments of thedisclosure;

FIG. 10 illustrates the statistical distribution of cascaded third-orderMZI's in the absence of coupler variations with respect to fabricationuncertainties, according to embodiments of the disclosure;

FIG. 11 illustrates the statistical distribution of MZI properties forthree waveguide widths, according to embodiments of the disclosure;

FIG. 12 illustrates the statistical distribution of an MZI filter withfour waveguide widths, according to embodiments of the disclosure;

FIG. 13 illustrates fabrication tolerance achieved using asymmetricwidths as well as heights, according to embodiments of the disclosure;

FIGS. 14A-14D illustrate unconventional cross-sections that arecompatible with CMOS-foundry processes, according to embodiments of thedisclosure;

FIG. 15 illustrates the performance of a filter, according toembodiments of the disclosure;

FIG. 16 illustrates performance of a filter, according to embodiments ofthe disclosure;

FIG. 17 illustrates performance of the filter, according to embodimentsof the disclosure;

FIG. 18 illustrates asymmetric widths where width and height variationsare independent, according to embodiments of the disclosure;

FIG. 19 illustrates a filter having asymmetric widths where width andheight variations are independent and each stage is correlated,according to embodiments of the disclosure;

FIG. 20 illustrates an embodiment where width and height variations ofevery stage are correlated, according to embodiments of the disclosure;

FIG. 21 illustrates an embodiment where width and height variations areindependent but are correlated for all stages, according to embodimentsof the disclosure;

FIG. 22 illustrates an embodiment where width and height variations areindependent but are correlated for all stages, according to embodimentsof the disclosure;

FIG. 23 illustrates the performance of a filter, according toembodiments of the disclosure;

FIG. 24 illustrates yield percentage, according to embodiments of thedisclosure;

FIG. 25 illustrates a simplified plan view of an example Mach-Zehnderinterferometer switch including a passive compensation structure,according to embodiments of the disclosure; and

FIG. 26 illustrates a simplified plan view of an example Mach-Zehnderinterferometer switch including a passive compensation structure,according to embodiments of the disclosure.

DETAILED DESCRIPTION

Some embodiments of the present disclosure relate to a passivecompensation structure for a Mach-Zehnder interferometer (MZI) filterthat improves the filter's ability to accommodate changes inmanufacturing tolerances and/or other perturbations. While the presentdisclosure can be useful for a wide variety of configurations, someembodiments of the disclosure are particularly useful for cascaded MZIfilters that are fabricated using silicon-based structures, as describedin more detail below.

For example, in some embodiments, an MZI filter includes a pair ofwaveguides that extend between a first and a second coupler section. Thefirst waveguide has a first continuous width along its length. Thesecond waveguide includes a tolerance compensation portion positionedbetween the first and the second coupler sections. The tolerancecompensation portion includes multiple waveguide sections, each having adifferent width, as explained in more detail below. The compensationportion can reduce a shift in frequency response of the MZI filter thatcan be caused by various perturbations, including variations inmanufacturing widths of the waveguides, manufacturing variations inthicknesses of the waveguides and variations in temperature. In furtherembodiments the compensation structure can be designed to reduce a shiftin frequency response of the MZI filter that can be caused by myriadperturbations while meeting a resonance requirement, as described inmore detail below.

In one example the tolerance compensation portion includes waveguidesections having three different widths, however other embodiments mayhave a lesser number or a greater number of widths. In this example, thetolerance compensation portion includes a first compensation portionhaving a second width, a second compensation portion having a thirdwidth and a third compensation portion having a fourth width, whereinthe fourth width is greater than the third width and the third width isgreater than the second width.

In another example the first waveguide can also have a compensationportion including multiple waveguide sections, each having differentwaveguide widths. In further examples, the compensation structure can bedesigned to compensate for a particular number of system perturbationsby having a quantity of waveguide widths that is greater than the numberof perturbations. In one embodiment the resonance requirement and anumber of system perturbations can be accommodated by designing thecompensation structure to have at least one more waveguide width thanthe number of system perturbations. For example in one embodiment a MZIfilter can be designed to have insensitivity to width variations and tohave a resonance at 1.55 um by having a compensation structure withthree different widths, while a compensation structure having twodifferent widths may be used to compensate for width variations only. Infurther examples, the degree to which the compensation structure cancompensate for a particular set of perturbations can be improved byincreasing the total number of different waveguide widths, as alsodescribed below.

In some embodiments, lengths and widths of the compensation structurecan be determined using one or more compensation equations. Morespecifically, the first and the second waveguides of the MZI filtersimultaneously satisfy:

${{m\lambda_{0}} = {L_{1}\left( {{n_{1}\left( \lambda_{0} \right)} - {\Sigma_{i}{n_{i}\left( \lambda_{0} \right)}\kappa_{i}}} \right)}}{v_{FSR} = \frac{c}{L_{1}\left( {n_{\mathcal{g}1} - {\sum_{i}{n_{\mathcal{g}i}\kappa_{i}}}} \right)}}{\frac{\partial n_{1}}{\partial X_{j}} = {\Sigma_{i}\kappa_{i}\frac{\partial n_{i}}{\partial X_{j}}}}{\frac{\partial^{2}n_{1}}{{\partial X_{j}}{\partial\omega}} = {\Sigma_{i}\kappa_{i}\frac{\partial^{2}n_{i}}{{\partial X_{j}}{\partial\omega}}}}{\frac{\partial^{2}n_{1}}{\partial X_{j}^{2}} = {\Sigma_{i}\kappa_{i}\frac{\partial^{2}n_{i}}{\partial X_{j}^{2}}}}$wherein:m=an integral multiple;λ₀=wavelength of light in first and second arms;L₁=reference length of first arm;L_(i)=length of i^(th) portion of second arm;κ_(i)=L_(i)/L₁;v_(FSR)=free spectral range;c=speed of light;X₁=waveguide width; andX₂=waveguide thickness.

In order to better appreciate the features and aspects of the presentdisclosure, further context for the disclosure is provided in thefollowing section by discussing one particular implementation of an MZIfilter that includes a passive compensation structure, according toembodiments of the disclosure. These embodiments are for explanatorypurposes only and other embodiments may be employed in other MZI-basedfilter devices. In some instances, embodiments of the disclosure areparticularly well suited for use with quantum computing circuits becauseof the intractability of using thermo-optic tuning for theseapplications.

FIG. 1A illustrates a simplified plan view of an example Mach-Zehnderinterferometer filter 100 including a passive compensation structure102, according to an embodiment of the disclosure. As shown in FIG. 1 ,MZI filter 100 includes a first waveguide 104 having a first length 106and extending from a first coupler section 108 to a second couplersection 110. First waveguide 104 has a constant first width 112 alongfirst length 106. A second waveguide 114 includes a compensation portion116 positioned between first coupler section 108 and second couplersection 110. Compensation portion 116 includes a first compensationsection 118 having a second width 120, a second compensation section 122having a third width 124 and a third compensation section 126 having afourth width 128. In some embodiments, fourth width 128 is greater thanthird width 124 and the third width is greater than second width 120. Insome embodiments, the width and length of each compensation portion canbe determined using one or more compensation equations, as described inmore detail below.

In some embodiments, compensation portion 116 is symmetric along secondwaveguide 114 and further includes a fourth compensation section 130having third width 124 and a fifth compensation section 132 havingsecond width 120. In further embodiments, compensation structure 102 mayalso include a compensation portion positioned within first waveguide104, as described in more detail below.

In various embodiments, one or more taper portions can be positionedin-between each compensation section to transition between differentwaveguide widths. More specifically, in some embodiments, a first taperportion 134 is positioned between first coupler section 108 and firstcompensation section 118 and transitions to second width 120. A secondtaper portion 136 can be positioned between first compensation section118 and second compensation section 122 and transitions from secondwidth 120 to third width 124. A third taper portion 138 can bepositioned between second compensation section 122 and thirdcompensation section 126 and transitions from third width 124 to fourthwidth 128. Similarly, a fourth taper portion 140 can be positionedbetween third compensation section 126 and fourth compensation section130 and transitions from fourth width 128 to third width 124. A fifthtaper portion 142 can be positioned between fourth compensation section130 and fifth compensation section 132 and transitions between thirdwidth 124 and second width 120. A sixth taper portion 144 can bepositioned between fifth compensation section 132 and second couplersection 110 and can transition from second waveguide width 120. In someembodiments, first waveguide 104 can also include one or more taperportions to transition widths between first coupler section 108 to firstwaveguide 104 and from the first waveguide to second coupler section110.

In some embodiments, each compensation section 118, 122, 126, 130, 132of compensation portion 116 may have a substantially constant width.More specifically, in some embodiments, first compensation section 118has a constant second width 120, second compensation section 122 has aconstant third width 124, third compensation section 126 has a constantfourth width 128, fourth compensation section 130 has a constant thirdwidth 124 and fifth compensation section 132 has a constant second width120.

In some embodiments, each compensation section can have a particularlength, as determined by one or more compensation equations, describedin more detail below. First compensation section 118 can have a secondlength 146, second compensation section 122 can have a third length 148,third compensation section 126 can have a fourth length 150, fourthcompensation section 130 can have a fifth length 152 and fifthcompensation section 132 can have a sixth length 154.

In some embodiments, first length 106 of first waveguide 104, length ofeach compensation section 118, 122, 126, 130 and 132, first width 112 offirst waveguide 104 and widths 120, 124, 128, 124, 120 of eachrespective compensation section 118, 122, 126, 130 and 132 ofcompensation structure 102 can be determined using one or morecompensation equations. More specifically, the first and the secondwaveguides of MZI filter 100 simultaneously satisfy:

${{m\lambda_{0}} = {L_{1}\left( {{n_{1}\left( \lambda_{0} \right)} - {\Sigma_{i}{n_{i}\left( \lambda_{0} \right)}\kappa_{i}}} \right)}}{v_{FSR} = \frac{c}{L_{1}\left( {n_{\mathcal{g}1} - {\sum_{i}{n_{\mathcal{g}i}\kappa_{i}}}} \right)}}{\frac{\partial n_{1}}{\partial X_{j}} = {\Sigma_{i}\kappa_{i}\frac{\partial n_{i}}{\partial X_{j}}}}{\frac{\partial^{2}n_{1}}{{\partial X_{j}}{\partial\omega}} = {\Sigma_{i}\kappa_{i}\frac{\partial^{2}n_{i}}{{\partial X_{j}}{\partial\omega}}}}{\frac{\partial^{2}n_{1}}{\partial X_{j}^{2}} = {\Sigma_{i}\kappa_{i}\frac{\partial^{2}n_{i}}{\partial X_{j}^{2}}}}$wherein:m=an integral multiple;λ₀=wavelength of light in first and second arms;L₁=reference length of first arm;λ₀=central wavelength of light in first and second arms;L_(i)=length of i^(th) portion of second arm;κ_(i)=L_(i)/L₁;v_(FSR)=free spectral range;c=speed of light;X₁=waveguide width; andX₂=waveguide thickness.

For example, in one embodiment, compensation equations can be used todefine a compensation structure for a pump-rejection filter for aquantum computer having the following parameters:

(i) 120 dB of pump rejection at wavelength λ₀=1.55 μm;

(ii) 25 mdB of signal loss; and

(iii) A free-spectral range (FSR) of 2.4 THz.

In other embodiments other suitable parameters can be defined for an MZIfilter, as appreciated by one of skill in the art.

FIG. 1B illustrates a simplified plan view of an example MZI filter 156including a passive compensation structure, according to an embodimentof the disclosure. As shown in FIG. 1B, MZI filter 156 is similar to MZIfilter 100 illustrated in FIG. 1A. However, in this embodiment, MZIfilter 156 includes a compensation portion positioned within eachwaveguide arm. More specifically, similar to MZI filter 100, MZI filter156 includes compensation portion 116 positioned within second waveguide114, however, MZI filter 156 also includes a second compensation portion158 positioned within first waveguide 160, as described in more detailbelow. As appreciated by one of skill in the art with the benefit ofthis disclosure any combination of compensation portions can be employedin an MZI filter and the compensation portions do not need to be thesame, or even have similar widths and/or lengths. As described in moredetail below, each compensation portion can be uniquely designedaccording to the compensation equations.

As shown in FIG. 1B, first waveguide 160 includes second compensationportion 158 that includes a plurality of compensation sections, eachhaving a width and a length as defined by a set of compensationequations, described in more detail herein. Second compensation portion158 is positioned between first coupler section 108 and second couplersection 110. Second compensation portion 158 includes a sixthcompensation section 164 having fifth width 174 and seventh length 176,a seventh compensation section 166 having sixth width 178 and a eighthlength 180, and an eighth compensation section 168 having fifth width174 and seventh length 176. As described above with regard to FIG. 1A,one or more taper portions can be positioned between waveguide sectionsof different widths to transition from one width to another width.

FIG. 1C illustrates a simplified model of an MZI filter 172 illustratinggeometrical parameters for a set of compensation equations. As shown inFIG. 1C, an MZI filter 172 is shown having two parallel waveguides, eachhaving a particular set of geometric parameters. In general, the phasedifference between the two waveguide arms is given by Equation (1).

$\begin{matrix}{{\phi(\omega)} = {{{k_{1}(\omega)}L_{1}} - {\sum\limits_{i = 2}^{n + 2}{{k_{i}(\omega)}L_{i}}}}} & \left( {{Eq}.1} \right)\end{matrix}$

In Equation (1), ω is the angular frequency of light, k_(i)(ω) is thewave number corresponding to the i^(th) waveguide width at angularfrequency ω, while L_(i) refers to the length of the i^(th) waveguide.Note that L_(i) could be negative, in which case it would mean that itis located on the other arm. In one example, L₁, L₂, L₄ are positivewhile L₃ is negative, then the two arm lengths are L₁+L₃ and L₂+L₄. Thesimplest case of this class of structures is when each arm has adifferent but uniform width.

Several constraints may be satisfied by the filter design. Firstly, thepump with central wavelength λ₀ can be situated at a transmissionminimum (since this is a pump-rejection filter). Therefore, theleft-hand side (LHS) of Equation (1) corresponds an integral multiple mof 2π at the center wavelength λ₀. Since k_(i)(λ₀)=2πn_(i)(λ₀)λ₀ ⁻¹, forEquation (2). In writing down the expression for the transmissionfunction, in some embodiments, it is proportional to sin²(Ø/2). Invarious embodiments Ø/2=mπ, or ϕ=2mπ.mλ ₀ =L ₁(n ₁(λ₀)−Σ_(i) n _(i)(λ₀)κ_(i))  (Eq. 2)

In Equation (2), κ_(i)=L_(i)/L₁. In addition, in some embodiments, itmay be desirable for the filter to possess a predetermined free-spectralrange (FSR). The free-spectral range can be obtained by settingϕ(ω₀+2πv_(FSR))−ϕ(ω₀)=±2π. Since the FSR may be smaller than the centralangular frequency ω₀, the various k_(i) can be expanded in a Taylorseries about k_(i)(ω₀), where dk_(i)/dω=v_(gi) ⁻¹=n_(gi)/c. Here n_(gi)refers to the group refractive index at the center wavelength λ₀. Thisyields Equation (3) for v_(FSR).

$\begin{matrix}{v_{FSR} = \frac{c}{L_{1}\left( {n_{\mathcal{g}1} - {\sum_{i}{n_{\mathcal{g}i}\kappa_{i}}}} \right)}} & \left( {{Eq}.3} \right)\end{matrix}$

To check the validity of Equation (3), a conventional MZI may beconsidered having arms of differing lengths L₁, L₂ but the same widths.This yields Equation (4) for v_(FSR).

$\begin{matrix}{v_{FSR} = {\frac{c}{n_{\mathcal{g}1}\left( {L_{1} - L_{2}} \right)} = \frac{c}{n_{\mathcal{g}1}\Delta L}}} & \left( {{Eq}.4} \right)\end{matrix}$

Next, constraints can be derived that make the system invariant tovarious sources of perturbation, X_(j). This can be achieved by setting

$\frac{\partial\phi}{\partial X_{j}} = 0.$A generic approach can be used in which N+1 waveguide widths are used tomitigate N sources of perturbation. In addition, the resonant wavelengthλ_(c) (defined as the location of the transmission minimum in this case)can be made invariant to perturbations as shown in Equation (5).

$\begin{matrix}{\frac{\partial n_{1}}{\partial X_{j}} = {\Sigma_{i}\kappa_{i}\frac{\partial n_{i}}{\partial X_{j}}}} & \left( {{Eq}.5} \right)\end{matrix}$

Equation (5) is generally valid for various sources of perturbation. Forexample, X₁≡w, where w is waveguide width and X₂≡h, where h is waveguidethickness. Additional sources of perturbation can be defined, i.e. X₃≡T,where T is temperature, etc. Each source of variation represents anadditional linear equation with unknowns κ_(i) for a given set of w_(i).

While Equation (5) adjusts the resonant wavelength λ_(c) (the wavelengthat which a transmission minimum is present) to be invariant toperturbation, it does not make the shape of the transmission curve nearthe minimum invariant. In some embodiments, this condition can beimposed by setting the derivative of ∂²ϕ/∂ω∂X_(j) to be constant.

In some embodiments, an additional condition can be imposed to mitigateN different sources of variation yielding Equation (6).

$\begin{matrix}{\frac{\partial^{2}n_{1}}{{\partial X_{j}}{\partial\omega}} = {\Sigma_{i}\kappa_{i}\frac{\partial^{2}n_{i}}{{\partial X_{j}}{\partial\omega}}}} & \left( {{Eq}.6} \right)\end{matrix}$

Equation (6) also represents a set of linear equations with unknownsκ_(i) for a given set of w_(i). In Equation (6), the order ofderivatives is swapped for the sake of convenience since ∂n_(i)/∂ω isreadily obtained from the effective-index dispersion of waveguides.Furthermore, Equations (2)-(3) can be reduced to a single equation withunknowns κ_(i) by dividing Equation (2) by Equation (3) as shown belowin Equations (7a) and (7b).

$\begin{matrix}{\frac{m\lambda_{0}}{c/{FSR}} = {\gamma = \frac{\left( {{n_{1}\left( \lambda_{0} \right)} - {\Sigma_{i}{n_{i}\left( \lambda_{0} \right)}\kappa_{i}}} \right)}{\left( {n_{\mathcal{g}1} - {\Sigma_{i}n_{{\mathcal{g}}i}\kappa_{i}}} \right)}}} & \left( {{{Eq}.7}a} \right)\end{matrix}$ $\begin{matrix}{{{\gamma n_{\mathcal{g}1}} - n_{1}} = {{\Sigma\kappa}_{i}\left( {{\gamma n_{{\mathcal{g}}i}} - n_{i}} \right)}} & \left( {{{Eq}.7}b} \right)\end{matrix}$

Equations (5), (6) and (7b) represent a set of 2N+1 linear equations inκ_(i) for 2N sources of perturbation or constraints. If Equation (6) isignored, then there are N+1 linear equations in κ_(i). Thus for apredefined set of N+1 waveguide widths, a solution is yielded byobtaining N+1 values of κ_(i). Since the various partial derivativesenumerated above are real, a solution to the above problem is generated.Negative values of κ_(i) are permitted since they represent that sectionbeing present in the ‘other’ arm. Thus, the above problem can thereforebe cast into a form MX=B as shown below in Equation (8).

$\begin{matrix}{M = \begin{bmatrix}{{\gamma n_{\mathcal{g}2}} - n_{2}} & {{\gamma n_{\mathcal{g}3}} - n_{3}} & \ldots & {{\gamma n_{{\mathcal{g}}({N + 2})}} - n_{N + 2}} \\\frac{\partial n_{2}}{\partial X_{1}} & \frac{\partial n_{3}}{\partial X_{1}} & \ldots & \frac{\partial n_{N + 2}}{\partial X_{1}} \\ \vdots & \vdots & \ldots & \vdots \\\frac{\partial n_{2}}{\partial X_{K}} & \frac{\partial n_{3}}{\partial X_{K}} & \ldots & \frac{\partial n_{N + 2}}{\partial X_{K}} \\\frac{\partial^{2}n_{2}}{{\partial\omega}{\partial X_{1}}} & \frac{\partial^{2}n_{3}}{{\partial\omega}{\partial X_{1}}} & \ldots & \frac{\partial^{2}n_{N + 2}}{{\partial\omega}{\partial X_{1}}} \\ \vdots & \vdots & \ldots & \vdots \\\frac{\partial^{2}n_{2}}{{\partial\omega}{\partial X_{K}}} & \frac{\partial^{2}n_{3}}{{\partial\omega}{\partial X_{K}}} & \ldots & \frac{\partial^{2}n_{N + 2}}{{\partial\omega}{\partial X_{K}}}\end{bmatrix}} & \end{matrix}$ $\begin{matrix}{X = \begin{bmatrix}\kappa_{2} \\\kappa_{3} \\ \vdots \\\kappa_{N + 2}\end{bmatrix}} & \end{matrix}$ $\begin{matrix}{B = \begin{bmatrix}{{\gamma n_{\mathcal{g}1}} - n_{1}} \\\frac{\partial n_{1}}{\partial X_{1}} \\ \vdots \\\frac{\partial n_{1}}{\partial X_{K}} \\ \vdots \\\frac{\partial^{2}n_{1}}{{\partial\omega}{\partial X_{1}}} \\ \vdots \\\frac{\partial^{2}n_{1}}{{\partial\omega}{\partial X_{K}}}\end{bmatrix}} & \left( {{Eq}.8} \right)\end{matrix}$

FIG. 2 illustrates a single stage of a three-waveguide Cascaded thirdorder MZI based filter 200 using a solution to Equation (8). Each stagecan be incoherently cascaded as shown in FIG. 3 that illustrates anincoherently cascaded third-order MZI filter 300 having four stages 305,310, 315, 320.

In some embodiments, it may be considered that the above set ofequations do not consider loss or extinction ratio thus it may bepossible that the obtained lengths from the above set of constraintsviolate the parameters of the extinction ratio.

In some embodiments, the use of more or less than N+1 waveguide widthscan be used. In either case, the problem is modified to an optimizationproblem, i.e. a solution to min(MX−B) is desirable.

In some embodiments, the transitions in waveguide widths may notconsidered because the waveguide widths may be marginally different andtherefore the transition lengths between these may not be relativelylarge, approximately 1 micron, in one embodiment. This can be relativelysmaller than the length of one of the arms, for example approximately100 microns, in one embodiment.

The discussion above disclosed an approach to make the MZI's tolerant tosources of perturbation. The next section discloses a design processincluding an approach to test the statistical performance of an MZIdevice.

The first step is to define the geometry of the device and obtainrefractive indices of waveguides as functions of w, h, T . . . and othervariables for various angular frequencies co. In some embodiments, thiscan be accomplished using commercial mode solvers. Upon obtaining thisinformation, it can be stored in the form of look-up tables. To simplifystoring the spectral dependencies, the refractive index data can be fitas follows and the coefficients n, ∂n/∂ω,∂²n/∂ω² can be stored yieldingEquation (9).

$\begin{matrix}{{n\left( {\omega,{X_{1}\ldots X_{N}}} \right)} = {{n\left( {\omega_{0},{X_{1}\ldots X_{N}}} \right)} + {\frac{\partial{n\left( {X_{1},{\ldots X_{n}}} \right)}}{\partial\omega}\left( {\omega - \omega_{0}} \right)} + {\frac{\partial^{2}{n\left( {X_{1},{\ldots X_{N}}} \right)}}{\partial\omega^{2}}\left( {\omega - \omega_{0}} \right)^{2}}}} & \left( {{Eq}.9} \right)\end{matrix}$

Equation (8) can then be solved to obtain various ratios κ_(i). If anexact solution cannot be obtained, variation of the central resonantwavelength Δλ_(c) can be minimized for given standard deviations inperturbation sources σ_(X) _(j) according to Equation (10).

$\begin{matrix}{{\Delta\lambda}_{c} = {\lambda_{0}\frac{\Sigma_{j}\sigma_{X_{j}}{❘{\frac{\partial n_{1}}{\partial x_{j}} - {\sum\limits_{i = 2}^{N + 2}{\kappa_{i}\frac{\partial n_{i}}{\partial X_{j}}}}}❘}}{n_{\mathcal{g}1} - {\sum\limits_{i = 2}^{N + 2}{\kappa_{i}n_{{\mathcal{g}}i}}}}}} & \left( {{Eq}.10} \right)\end{matrix}$

The value of L₁ can be determined using Equation (3). The second MZI inthe third-order MZI will can possess L′₁=2L₁ but the same values ofκ_(i). Using the obtained values of L_(i), the values of t₁, t₂, t₃ maybe optimized as well as a number of stages N to meet the specificationsof extinction ratio, transmission loss and extinction bandwidth. In someembodiments, extinction bandwidth (BW) may be larger than the centralwavelength shift Δλ_(c), e.g. BW>>Δλ_(c). A Monte-Carlo analysis of thesystem can be performed by repeating a relatively large number (R_(N))of random simulations. The sampling can be conducted with knowledge ofcorrelations in a representative fabrication process. In someembodiments, the process can be repeated until a favorable yield isobtained.

In some embodiments, numerical methods can be used to develop a MZIfilter. The output of a filter can obtained using transfer matrices. Acascaded third-order filter can include directional couplers and thepropagation of light in the two arms. A filter can be defined to bethird-order when two asymmetric MZI's of differential length ΔL and 2ΔLare cascaded coherently. The transfer matrices for directional couplersand MZI arms are shown in Equation (11).

$\begin{matrix}{{M_{c} = \begin{bmatrix}t & {- {jK}} \\{- {jK}} & t\end{bmatrix}},{M_{pm} = {\alpha^{1/2}\begin{bmatrix}{e^{{- j}\phi_{m}}\alpha_{m}} & 0 \\0 & 1\end{bmatrix}}}} & \left( {{Eq}.11} \right)\end{matrix}$

In Equation (11), |t|² is the transmission coefficient of thedirectional coupler. Notably, K=√{square root over (1−|t|²)} while Ø_(r)(r=1, 2) corresponds to the differential phase in each of the twoasymmetric MZI's that constitute a cascaded third-order filter.α_(m)=e^(−rα′ΔL/2) correspond to the additional losses that accrue dueto the differential length in each MZI, while α=e^(−rα′L/2) is thecommon absorption experienced by the nominal length L of the MZI arms.For the general multi-waveguide case, L=min(L₁, Σ_(i)κ_(i)L₁) andΔL=|L₁−Σ_(i)κ_(i)L₁|. Note that α′ is the absorption coefficient inunits of 1/meter.

Upon utilizing the above transfer matrices the following expressions forthe elements H_(mk) of the overall transfer matrix of the cascadedthird-order filter was obtained. A single third-order filter can bedefined by three couplers with corresponding parameters t₁, t₂, t₃ andtwo phase and absorption terms Ø_(r), α_(r), where r=1, 2 as shown inEquations (12a), (12b), (12c) and (12d).H ₁₁(ω)=α[−K ₁(ω)(t ₂(ω)K ₃(ω)+α₂ K ₂(ω)t ₃(ω)e ^(−jϕ) ² ^((ω)))−α₁ t₁(ω)e ^(−jϕ) ¹ ^((ω))(K ₂(ω)K ₃(ω)−α₂ t(ω)t ₃(ω)e ^(−j) ² ^((ω)))]  (Eq.12a)H ₁₂(ω)=α[−jt ₁(ω)(K ₃(ω)t ₂(ω)+α₂ K ₂(ω)t ₃(ω)e ^(−jϕ) ² ^((ω)))+jα₁(ω)K ₁(ω)e ^(−jϕ) ¹ ^((ω))(K ₂(ω)K ₃(ω)−α₂ t(ω)t ₃(ω)e ^(−jϕ) ²^((ω)))]  (Eq. 12b)H ₂₁(ω)=α[t ₁ −jK ₁(ω)(t ₂(ω)t ₃(ω)+α₂ K ₂(ω)K ₃(ω)e ^(−jϕ) ² ^((ω)))−jα₁ t ₁(ω)e ^(−jϕ) ¹ ^((ω))(K ₂(ω)t ₃(ω)−α₂ t ₂(ω)K ₃(ω)e ^(−jϕ) ²^((ω)))]  (Eq. 12c)H ₂₂(ω)=α[t ₁(ω)(t ₂(ω)t ₃(ω)+α₂ K ₂(ω)K ₃(ω)e ^(−jϕ) ² ^((ω)))+jα ₁ k₁(ω)e ^(−jϕ) ¹ ^((ω))(K ₂(ω)t ₃(ω)+α₂ K ₃(ω)t ₂(ω)e ^(−jϕ) ²^((ω)))]  (Eq. 12d)

The validity Equations (12a)-(12d) can be shown by verifying that|H_(qp)(ω)|²+|H_(pp)(ω)|²=1 for q, p=1, 2 under conditions of no loss(i.e. α′=0). This relates to the conservation of energy. Thetransmission loss and pump-rejection ratios can be calculated inEquations (13a) and (13b), respectively.

$\begin{matrix}{t_{loss} = {10{\log_{10}\left( \frac{\int_{- \infty}^{\infty}{{❘{E_{{out},1}(\omega)}❘}^{2}\left( {{I_{s}(\omega)} + {I_{i}(\omega)}} \right)d\omega}}{\int_{- \infty}^{\infty}{\left\lbrack {{I_{s}(\omega)} + {I_{i}(\omega)}} \right\rbrack d\omega}} \right)}{dB}}} & \left( {{{Eq}.13}a} \right)\end{matrix}$ $\begin{matrix}{t_{pump} = {1000 \times 10{\log_{10}\left( \frac{\int_{- \infty}^{\infty}{{❘{E_{{out},2}(\omega)}❘}^{2}{I_{p}(\omega)}d\omega}}{\int_{- \infty}^{\infty}{{I_{p}(\omega)}d\omega}} \right)}{mdB}}} & \left( {{{Eq}.13}b} \right)\end{matrix}$

In these embodiments the waveguides considered are silicon-on-insulator(SOI) strip waveguides, however other embodiments can use differentconfigurations. The material dispersion can be based on the Palik modelat room temperature. The dispersion of the effective index can be fitaccording to Equation (9). In this embodiment the center wavelengthλ₀=2πc/ω₀=1.55 μm. The obtained coefficients are plotted in FIG. 4showing the effective index parameters as a function of waveguide widthand height for a silicon-on-insulator waveguide. The obtained effectiveindex is fit to Equation (9).

A full parameter sweep of the refractive index over angular frequencyco, waveguide width (w) and thickness (h) is performed. In FIGS. 5A-5B,the derivatives are plotted with respect to w, h of the effective indexat the wavelength λ₀=1.55 μm.

In FIGS. 5A and 5B, the derivative ∂n/∂w is plotted. The value isinvariant with thickness but changes dramatically with width. Thisindicates that waveguide width variations can be mitigated using thisapproach. On the other hand, while ∂n/∂h does vary with width, it onlydoes so mildly; it varies more with regard to thickness. The magnitudeof change is about four times larger than ∂n/∂w. In some embodiments,standard deviations σ_(h) of the thickness tend be smaller than those ofthe widths (see Table. 1 showing parameters of simulations), whichreduces their impact.

TABLE 1 Parameters used for simulations. Parameter Value Standarddeviation of 3 nm⁵ waveguide width (σ_(ω)) Standard deviation of 0.5 nm⁶waveguide height (σ_(h)) Material index Palik (from Lumerical)Temperature (T) 300 K Absorption coefficient (α′) 0.3 dBcm⁻¹ or 7.5 m⁻¹Transmission coefficients 0.5, 0.75, 0.93 (|t₁(ω₀)|², |t₂(ω₀)|²,|t₃(ω₀)|²) Number of stages 4 Pump, signal and idler distributionsGaussian with 5 GHz e⁻² bandwidth Correlations ω, h for each third-orderMZI stage uncorrelated.

Due to the relative invariance of ∂n/∂h, with respect to w, the strategyof using multiple waveguide widths to mitigate variation in thisparameter may not be very efficacious for particular applications. Inprinciple, a solution is possible but the lengths of arms obtained turnout to be in the range of centimeters which can be too large for someapplications. Therefore, in some applications that may benefit fromsmall filter sizes, it would be beneficial to reduce the values ofσ_(h).

The coupling coefficients of the directional couplers can be determinedby obtaining the even and odd modes of the coupled waveguide system. Thecoupling length can then be determined according to Equation (14).

$\begin{matrix}{{t_{m}(\omega)} = {\sin\left\lbrack {\frac{\Delta{n(\omega)}\lambda_{0}}{\Delta{n\left( \omega_{0} \right)}\lambda}{\sin^{- 1}\left( {t_{m}\left( \omega_{0} \right)} \right)}} \right\rbrack}} & \left( {{Eq}.14} \right)\end{matrix}$

The statistical performance of standard cascaded third-order filters isexamined to estimate the yield for such devices. In this approach aMonte-Carlo calculation was employed. Waveguide widths and thicknesseswere chosen at random and their effective indices are obtained from thepreviously generated look-up tables. Similarly, the effective super-modeindices of the couplers are obtained. The coupling coefficients are thencalculated using Equation (14) and the parameters from Table 1 are used.The standard deviations for waveguide width σ_(w)=3 nm and thicknessσ_(h)=0.5 nm are plotted in FIG. 6 showing statistical distributions ofeffective index n_(eff).

In this embodiment the entire dispersion curve has been shifted. Thedistribution of effective indices is slightly asymmetric. Therefore, inassuming a 3 nanometer waveguide width standard deviation and 0.5nanometer standard deviation in thickness, this example evaluatesvariations more germane to die-to-die or intra-die variations.Therefore, a relevant parameter may be the critical dimension uniformity(CDU).

The overall performance for a N=4 stage, incoherently cascadedthird-order filter can then be obtained. The design described above hadthe goal of meeting the specifications fora pump rejection filter, thatcan be, in one example, 120 dB of rejection and 50 mdB of loss. However,from FIG. 7 that shows the statistical behavior of cascaded third-orderMZI's without mitigation mechanisms, it can be seen that the meanrejection ratio has shifted to approximately 60 dB and the meanabsorption coefficient has shifted to approximately 1800 mdB, howeverthese may have different values in other embodiments.

In one embodiment, a fabrication tolerant MZI design uses asymmetricwidths for each MZI arm. In this particular embodiment it is desired tomitigate variations to both thickness (h) and width (w), so the quantityin Equation (10) is minimized. The results are plotted in FIG. 8 thatillustrates designs to minimize susceptibility to fabrication errors.The minimization procedure yields a value of Δλ_(c)≈700 pm at variousvalues of κ for varying values of w₁ and h=220 nm. Incidentally, theminimization yields ∂n₁/∂w−∂n₂/∂w=0, while being at the mercy ofσ_(h)|∂n₁/∂h−∂n₂/∂h|. Therefore, in some embodiments, σ_(h) should bereduced.

As shown in FIG. 9 , the statistical distribution of cascadedthird-order MZI's with asymmetric widths w₁=500 nm and w₂=540 nm, h=220nm are illustrated. In some embodiments, this value can be reduced byincreasing the height of the waveguides. For instance at h=245.5 nm,Δλ_(c)≈580 pm. However, it also appears that using thicker waveguides insome embodiments causes the transmission loss to increase due todispersion. Therefore, over engineering this aspect of the system maynot be worthwhile for some embodiments. Using such a configuration, FIG.9 illustrates the performance for N=4 incoherently cascaded third-orderfilters. An improvement in performance compared to that depicted in FIG.7 is evident with the mean rejection ratio shifting to 110 dB and meanloss shifting to 188 mdB.

Furthermore, if coupler variations with respect to fabricationuncertainties (simply referred to as coupler variations henceforth) areignored, then the performance is shown in FIG. 10 illustrating thestatistical distribution of cascaded third-order MZI's in the absence ofcoupler variations with respect to fabrication uncertainties. Therejection ratio shifts to 154 dB, while the transmission loss changes to165 mdB. In some embodiments, this can indicate that coupler variationspredominantly produce vertical movements in the spectral response whilethe index changes produce mainly horizontal shifts. Horizontal shiftsaffect both rejection ratio and transmission loss, while vertical shiftspredominantly affect rejection ratios.

In some embodiments, while using asymmetric arms can make|∂n₁/∂w−∂n₂/∂w|=0, it may not correlate to a transmission minimumlocated at λ₀=1.55 μm. In the above embodiments, it is fortuitous thatfor κ+δκ, the above resonance condition is satisfied. Here, δκ is arelatively small amount of adjustment imparted to κ. Therefore, theremay be a residual error of −δκa∂n₂/∂w, which is may be undesirable.However, if two additional waveguide widths are used (i.e. w₂, w₃), thensome embodiments may have improved results. This is demonstrated in FIG.11 , where one of the arms contains two widths of 0.5, 0.66 microns.More specifically, FIG. 11 illustrates the statistical distribution ofMZI properties for three waveguide widths L₁=22.96 μm, m=58,κ_(i)=[1,4.2805,−4.6974] and w_(i)=[0.5,0.56,0.66] microns. In thisembodiment, the design can be constrained to satisfy a condition fortransmission minimum at λ₀ (Equation (1)), FSR (Equation (3)) andinsensitivity to width variations (Equation (5)).

In FIG. 11 , coupler variations are neglected, building on the resultsfrom FIG. 10 . As can be seen, there is an additional 10 dB improvementin rejection ratio, while an improvement in transmission loss byapproximately 70 mdB. While thickness variations may not be mitigatedusing this approach since ∂n/∂h is not a function of w, the additionalconstraint of having ∂²Ø∂w∂ω=0 may be included, which yields theperformance in FIG. 12 showing the statistical distribution of an MZIfilter with four waveguide widths. L₁=25.51 microns, κ_(i)=[1, 4.1464,−4.5875, 0.1662] and w_(i)=[0.5, 0.56, 0.66, 0.76] microns. The shape ofthe distribution appears to change, although improvements in mean valuesdo not appear to occur.

In principle, compensation for perturbations in w, h can besimultaneously achieved by choosing arms with different w, h as shown inEquation (15).

$\begin{matrix}{\frac{\partial n_{1}}{\partial w}{❘_{w_{1},h_{1}}{= \frac{\partial n_{2}}{\partial w}}❘}_{w_{2},h_{2}}} & \left( {{Eq}.15} \right)\end{matrix}$ $\begin{matrix}{\frac{\partial n_{1}}{\partial h}{❘_{w_{1},h_{1}}{= \frac{\partial n_{2}}{\partial h}}❘}_{w_{2},h_{2}}} & \end{matrix}$

This results in a value of Δλ_(c)=26 pm. The results are plotted in FIG.13 illustrating fabrication tolerance achieved using asymmetric widthsas well as heights. w₁=500 nm, w₂=535 nm, h₁=220 nanometers and h₂=245nanometers. The average pump rejection shifts to 174 dB and the averageloss is 125 mdB, which is smaller compared to the case when onlyasymmetric widths without coupler variations (FIG. 10 ) are considered.Here, too the effect of coupler variations have been ignored. Themarginal increase in absorption relative to FIGS. 11 and 12 is that theconstraint of fixing A is not satisfied. In some embodiments, waveguidegeometries that effectively enable different heights (such as ribwaveguides) can be used. Furthermore, some embodiments can use bothdifferent heights and multiple widths to further improve performance.

While obtaining different thicknesses can be challenging in someembodiments, there may be ways to accomplish this by usingunconventional cross-sections that are compatible with currentCMOS-foundry processes, as shown in FIGS. 14A-14D. In one exampleembodiment shown in FIG. 14A, a conventional strip waveguide is shown.In FIG. 14B, a cross-section which has an additional silicon nitride orsilicon layer on top of the SOI strip waveguide that modifies theeffective height is shown. FIG. 14C illustrates a rib waveguide and FIG.14D illustrates a modified rib waveguide showing two other embodimentsthat offer height changes.

The effect of coupler dispersion and insertion loss on the system cannow be considered. All the systems are assumed to possess thethree-waveguide design from FIG. 11 , however other embodiments may haveother configurations. In the first embodiment illustrated in FIG. 15 ,the coupler dispersion is retained while waveguide loss is reduced to0.1 dB/cm. FIG. 15 illustrates performance of the filter when the lossis reduced to 0.1 dB/cm and with couplers robust to fabricationvariations but with varying transmission with respect to frequency.

With this improvement, the transmission loss has reduced to 93 mdB,while the pump rejection has been minimally altered. On the contrary,when the loss is maintained at 0.3 dB/cm but the couplers arefab-tolerant and also not dispersive, the transmission loss falls belowthe 50 mdB level as shown in FIG. 16 . FIG. 16 illustrates performanceof the filter when the loss is 0.3 dB/cm with couplers that are robustto fabrication variations and also with constant transmission withrespect to frequency.

Thus, in order to meet device specifications, in some embodiments, thecouplers may be fab-tolerant and broadband. When the loss is alsoreduced to 0.1 dB/cm with couplers robust to fabrication and also withconstant transmission coefficients with respect to frequency,transmission losses reduce to 28 mdB as can be seen in FIG. 17 . FIG. 17illustrates performance of the filter when loss is reduced to 0.1 dB/cmwith couplers robust to fabrication variations and also with constanttransmission with respect to frequency. In order to reduce transmissionloss values below 25 mdB, the number of stages may be reduced to three,although this may also reduce the mean pump rejection ratio. Therefore,in order to further improve the yield, improved process control orreduced values of σ_(w), σ_(h) may be needed.

As described above, the variations of width and thickness were treatedas independent random variables and each stage was assumed to varyindependently. In this section the case when the width and heightvariations are uncorrelated but all stages are well-correlated isevaluated. When the correlation between each stage increases, the spreadin performance increases as shown in FIG. 18 . FIG. 18 illustratesasymmetric widths where width and height variations are independent andevery stage is correlated with a loss of 0.3 dB/cm.

However, in some embodiments, if the couplers are made insensitive tofabrication, then the performance improves as seen in FIG. 19 . FIG. 19shows an embodiment having asymmetric widths where width and heightvariations are independent and each stage is correlated. Couplers areconsidered fabrication tolerant and the loss is 0.3 dB/cm. Using threeor four waveguide widths, as was the case in FIGS. 11 and 12 , improvesperformance even more, bringing elements close to specifications in FIG.20 . FIG. 20 illustrates an embodiment where width and height variationsof every stage are correlated. In additional to mitigating couplervariations, one embodiment uses three waveguide widths. The waveguideloss assumed in this embodiment is 0.3 dB/cm.

If develop broadband couplers are developed while maintaining loss at0.3 dB/cm, the performance improvement is line with trends in theprevious embodiments, as shown in FIG. 21 . FIG. 21 illustrates anembodiment where width and height variations are independent but arecorrelated for all stages. The structure uses three waveguide widths andfab-tolerant and broadband couplers and the insertion loss is 0.3 dB/cm.As illustrated in FIG. 22 , the loss is reduced to 0.1 dB/cm, whichbrings the performance to similar levels as shown in FIG. 17 .

More specifically, even when the correlations are not favorable, thedevices have comparable yield. FIG. 22 illustrates an embodiment wherewidth and height variations are independent but are correlated for allstages. The structure uses three waveguide widths, fab-tolerant andbroadband couplers as well as a reduced insertion loss of 0.1 dB/cm.

Serial improvements are summarized that can be achieved for variousdesign improvements shown in Table 3. In some embodiments, broadband,fabrication insensitive couplers enable the system to meet performancespecifications. In further embodiments, reducing waveguide losseson-chip may help improve the performance and yield. In additionembodiments having σ_(w)<3 nanometers and σ_(h)<0.5 nanometers may beused.

Table 3 summarizes different embodiments that may have reducedperformance and also identifies various strategies that couldpotentially address the performance. Each point labelled (i)-(iv) inTable 3 is discussed in more detail below.

(i) In some embodiments, the use of asymmetric arm widths may achievetuning-free operation of cascaded third-order filters. Use of three orfour waveguide widths helps achieve pinning the transmission minimum andalso compensates ∂²n/∂w∂ω.

(ii) In some embodiments, the use of multi-waveguide sections canmitigate many sources of variation but due to the invariance of do/ah tow, this approach may need long device lengths to mitigate thicknessvariations. In principle, using different waveguide heights can alsoaddress thickness variation issues, although this may not be aCMOS-foundry compatible process. Some embodiments may use unconventionalwaveguide geometries to effectively engineer a height difference.

TABLE 2 Mean Pump Mean μ_(loss) + rejection μ − ≥120 dB Transmissionσ_(loss) ≤25 mdB Design μpump (dB) σ_(pump)(dB) (%) loss (mdB) (mdB) (%)Standard third-order, 60 44 0 1830 3074 0 4 stage MZI, 0.3 dB/cmAsymmetric widths, 0.3 dB/cm 110 92 28 188 280 0 Asymmetric widths androbust 154 130 87 164 237 0 couplers, 0.3 dB/cm Multiple widths, robustcouplers, 163 97.4 99.4 110 155.1 0 0.3 dB/cm Standard third-order, 16398.2 99.43 44 80 2.2 4 stage MZI, 0.3 dB/cm Multiple widths, robustcouplers 163 143.5 97.1 93 144.1 0 and 0.1 dB/cm loss Multiple widths,robust and 163 141.5 97.4 28 67 67.3 broadband couplers and 0.1 dB/cmloss Multiple widths, robust couplers 122 104 55.1 22 56 76.2 and 0.1dB/cm loss, 3 stages σ_(w) = 1 nm, σ_(h) = 0.25 nm and Multiple 188 175100 10.5 12.5 99.7 widths, robust, broadband couplers, 0.1 dB/cm, 4stages σ_(w) = 1 nm, σ_(h) = 0.25 nm and Multiple 141 131 98.5 7.8 8.8100 widths, robust, broadband couplers, 0.1 dB/cm, 3 stages Asymmetricwidths, correlated stage 111 73.26 36 210 609 0 variations, 0.3 dB/cmAsymmetric widths, robust couplers 148 102 69 178 329.3 0 and correlatedstage variations, 0.3 dB/cm Multiple widths, robust couplers and 164 12378 105 177 0 correlated stage variations, 0.3 dB/cm Multiple widths,robust and 163 120.75 80 44 103 21 broadband couplers and correlatedstage variations, 0.3 dB/cm Multiple widths, robust and 163 120 78 28 9579 broadband couplers and correlated stage variations, 0.1 dB/cm σ_(w) =1 nm, σ_(h) = 0.25 nm and Multiple 187 163 99.2 10.5 13.85 99.4 widths,robust, broadband couplers, 0.1 dB/cm, correlated σ_(w) = 1 nm, σ_(h) =0.25 nm and Multiple 141 124 91 7.8 10.38 99.6 widths, robust, broadbandcouplers, 0.1 dB/cm, 3 stages

TABLE 3 Problem Reason Value Strategy (i) Resonance shift Multiplewaveguide widths (ii) Sensitivity Equal Si-SiO₂-Si, Si-SiN-Si to heightheights waveguides (iii) Transmission Dispersion 62 mdB Broadbandcouplers loss in couplers (iv) Bandwidth of Roll-off 2 nm at Reducedwaveguide loss filter −150 dB Additional stages or alternatearchitectures

As shown in FIG. 23 , in some embodiments, reducing σ_(W) to 1 nanometerand σ_(h) to 0.25 nanometer from 3 and 0.5 nanometer respectivelyenables the specifications to comfortably meet the goals.

(iii) In some embodiments, the role of coupler dispersion and variationswith fabrication may be important. Designing couplers that are morebroadband and insensitive to fabrication variations may be needed tomake a filter robust to perturbations.

(iv) In some embodiments, to meet specifications, loss may reachapproximately 0.1 dB/cm. This may enable specifications to be exceededby adding further cascaded third-order MZI filter stages. In furtherembodiments, using three stages may meet rejection ratio targets whilekeeping losses below the 25 mdB level.

(v) In some embodiments, further improvement of fabrication tolerancesto σ_(w)<<3 nm and σ_(h)<<0.5 nanometer may improve the mean pumprejection to 188 dB and average loss to 10.54 mdB for a four stagecascaded third-order MZI as is seen in FIGS. 23A and 23B.

FIG. 24 illustrates yield percentage of loss <25 mdB and rejectionratio >120 db. Variations are correlated, with insertion loss of 0.1dB/cm, broadband and fab-tolerant couplers and multiple waveguide widtharms.

Although MZI filter 100 (see FIG. 1 ) is described and illustrated asone particular type of MZI-based photonic device, a person of skill inthe art with the benefit of this disclosure will appreciate thatcompensation structures as described above are suitable for use withmyriad other MZI-based photonic devices. For example, in someembodiments the MZI passive compensation structures disclosed herein canbe implemented in MZI-based photonic switching devices, as described inmore detail below.

FIGS. 25 and 26 show example MZI-based photonic switches 2500 and 2600,respectively, that include one or more variable phase-shifters and canalso include one or more compensation structures. Photonic switches 2500and 2600 are similar to MZI filter 100 (see FIG. 1 ), each having twoparallel waveguides (2510 a, 2510 b in FIGS. 25, and 2610 a and 2610 bin FIG. 26 ), however photonic switches 2500 and 2600 each include oneor more phase shifters (2505 a, 2505 b, 2505 c in FIGS. 25 and 2605 inFIG. 26 ) disposed in one or more waveguides of each photonic switch.Phase-shifters (2505 a, 2505 b, 2505 c in FIGS. 25 and 2605 in FIG. 26 )can be implemented a number of ways in integrated photonic circuits andcan provide control over the relative phases imparted to the opticalfield in each waveguide. In some embodiments, variable phase-shifterscan be implemented using thermo-optical switches.

In some embodiments thermo-optical switches can use resistive elementsfabricated on a surface of the photonic device. Employing thethermo-optical effect in these devices can provide a change of therefractive index n by raising the temperature of the waveguide by anamount of the order of 10⁻⁵K. One of skill in the art having had thebenefit of this disclosure will understand that any effect that changesthe refractive index of a portion of the waveguide can be used togenerate a variable, electrically tunable, phase shift. For example,some embodiments can use beam splitters based on any material thatsupports an electro-optic effect. In some embodiments so-called χ⁽²⁾ andχ⁽³⁾ materials can be used such as, for example, lithium niobate, BBO,KTP, BTO, and the like and even doped semiconductors such as silicon,germanium, and the like.

In some embodiments, switches with variable transmissivity and arbitraryphase relationships between output ports can also be achieved bycombining directional couplings (e.g., directional couplings 2515 a,2515 b in FIGS. 25 and 2615 a, 2615 b in FIG. 26 ), and one or morevariable phase-shifters (e.g., phase-shifters 2505 a, 2505 b, 2505 c inFIGS. 25 and 2605 in FIG. 26 ) within each photonic switch. Accordingly,complete (e.g., analog or digital) control over the relative phase andamplitude of the two output ports can be achieved by varying the phasesimparted by phase shifters (2505 a, 2505 b, 2505 c in FIGS. 25 and 2605in FIG. 26 ). FIG. 26 illustrates a slightly simpler example of aMZI-based photonic switch that allows for variable transmissivitybetween ports 2620 a and 2620 b by varying a phase imparted by phaseshifter 2605.

In some embodiments one or more compensation structures can beimplemented within MZI-based photonic switches 2500,2600 usingcompensation equations similar to those described above with regard toMZI filter 100 (see FIG. 1 ). More specifically, the compensationequations can be used to determine a width and a length of eachcompensation portion that can be used to reduce a shift in frequencyresponse caused by various perturbations, including variations inmanufacturing widths of the waveguides, manufacturing variations inthicknesses of the waveguides and variations in temperature. Similar tothe compensation structures described for MZI filter 100 (see FIG. 1 ),compensation structures can be employed in one or more waveguides (2510a, 2510 b in FIGS. 25, and 2610 a and 2610 b in FIG. 26 ), and eachcompensation structure can each have a quantity of waveguide widths thatis greater than the number of perturbations, however the governingequations may be different for an MZI-based photonic switch embodiment,as described in more detail below.

The phase relationship in an MZI-based photonic switch embodiment may bedescribed as follows. The first two terms can be the same as MZI filter100 (see FIG. 1 ), however a third term corresponding to a sum ofvarious index changes, Δn_(j), weighted by various overlap integralsΓ_(j), can be added, as described by Equations (16) and (17).

$\begin{matrix}{\frac{\left( {{2m} + 1} \right)\lambda_{0}}{2} = {{{n_{1}\left( \omega_{0} \right)}L_{1}} - {\Sigma_{i}\kappa_{i}L_{1}{n_{i}\left( \omega_{0} \right)}} + {\Sigma_{j}{\Gamma_{j}\left( \omega_{0} \right)}\Delta{n_{j}\left( \omega_{0} \right)}L_{1}}}} & \left( {{Eq}.16} \right)\end{matrix}$ $\begin{matrix}{v_{FSR} = \frac{c}{L_{1}\left( {n_{\mathcal{g}1} - {\Sigma_{i}n_{{\mathcal{g}}i}\kappa_{i}}} \right)}} & \left( {{Eq}.17} \right)\end{matrix}$The corresponding compensation equation for the case of an MZI-basedphotonic switch which requires invariance to width can be described byEquation (18).

$\begin{matrix}{\frac{\partial\phi_{j}}{\partial w} = {{L_{1}\left( {\frac{\partial n_{1}}{\partial w} - \frac{\Sigma_{i}\kappa_{i}{\partial n_{i}}}{\partial w} + {\frac{\Sigma_{j}{\partial\Gamma_{j}}}{\partial w}\Delta n_{j}}} \right)} = 0}} & \left( {{Eq}.18} \right)\end{matrix}$Equation (18) can be reduced to Equation (19).

$\begin{matrix}{\frac{\Sigma_{i}\kappa_{i}{\partial n_{i}}}{\partial w} = {{\Sigma_{j}\frac{\partial\Gamma_{j}}{\partial w}\Delta n_{j}} + \frac{\partial n_{1}}{\partial w}}} & \left( {{Eq}.19} \right)\end{matrix}$In some embodiments, Equations (20) through (22) can be used to accountfor compensation of higher-order derivatives.

$\begin{matrix}{\frac{\Sigma_{i}\kappa_{i}{\partial^{2}n_{i}}}{{\partial w}{\partial\omega}} = {{\Sigma_{j}{\frac{\partial}{\partial\omega}\left\lbrack {\frac{\partial\Gamma_{j}}{\partial w}\Delta n_{j}} \right\rbrack}} + \frac{\partial^{2}n_{1}}{{\partial w}{\partial\omega}}}} & \left( {{Eq}.20} \right)\end{matrix}$ $\begin{matrix}{\frac{\Sigma_{i}\kappa_{i}{\partial^{2}n_{i}}}{\partial w^{2}} = {{\Sigma_{j}\left\lbrack {\frac{\partial^{2}\Gamma_{j}}{\partial w^{2}}\Delta n_{j}} \right\rbrack} + \frac{\partial^{2}n_{1}}{\partial w^{2}}}} & \left( {{Eq}.21} \right)\end{matrix}$ $\begin{matrix}{\frac{\left( {{n_{1}\left( \omega_{0} \right)} - {\Sigma_{i}{n_{i}\left( \omega_{0} \right)}\kappa_{i}} + {\Sigma_{j}{\Gamma_{j}\left( \omega_{0} \right)}\Delta{n_{j}\left( \omega_{0} \right)}}} \right)}{\left( {n_{\mathcal{g}1} - {\Sigma_{i}n_{{\mathcal{g}}i}\kappa_{i}}} \right)} = {\gamma = \frac{\left( {m + \frac{1}{2}} \right)\lambda_{0}}{c/{FSR}}}} & \left( {{Eq}.22} \right)\end{matrix}$Generalizing to arbitrary perturbations X_(k), the set of compensationequations for MX=B can be described by Equations (23) through (25).

$\begin{matrix}{M = \begin{bmatrix}{{\gamma n_{\mathcal{g}2}} - n_{2}} & {{\gamma n_{\mathcal{g}3}} - n_{3}} & \ldots & {{\gamma n_{{\mathcal{g}}({N + 2})}} - n_{N + 2}} \\\frac{\partial n_{2}}{\partial X_{1}} & \frac{\partial n_{3}}{\partial X_{1}} & \ldots & \frac{\partial n_{N + 2}}{\partial X_{1}} \\ \vdots & \vdots & \ldots & \vdots \\\frac{\partial n_{2}}{\partial X_{K}} & \frac{\partial n_{3}}{\partial X_{K}} & \ldots & \frac{\partial n_{N + 2}}{\partial X_{K}} \\\frac{\partial n_{2}}{{\partial\omega}{\partial X_{1}}} & \frac{\partial n_{3}}{{\partial\omega}{\partial X_{1}}} & \ldots & \frac{\partial n_{N + 2}}{{\partial\omega}{\partial X_{1}}} \\ \vdots & \vdots & \ldots & \vdots \\\frac{\partial n_{2}}{{\partial\omega}{\partial X_{K}}} & \frac{\partial n_{3}}{{\partial\omega}{\partial X_{K}}} & \ldots & \frac{\partial n_{N + 2}}{{\partial\omega}{\partial X_{K}}}\end{bmatrix}} & \left( {{Eq}.23} \right)\end{matrix}$ $\begin{matrix}{X = \begin{bmatrix}\kappa_{2} \\\kappa_{3} \\ \vdots \\\kappa_{N + 2}\end{bmatrix}} & \left( {{Eq}.24} \right)\end{matrix}$ $\begin{matrix}{B = \begin{bmatrix}{{\gamma n_{\mathcal{g}1}} - n_{1} - {\Sigma_{j}{\Gamma_{j}\left( \omega_{0} \right)}\Delta{n_{j}\left( \omega_{0} \right)}}} \\{\frac{\partial n_{1}}{\partial X_{1}} + {\Sigma_{j}\frac{\partial\Gamma_{j}}{\partial X_{1}}\Delta n_{j}}} \\ \vdots \\{\frac{\partial n_{1}}{\partial X_{K}} + {\Sigma_{j}\frac{\partial\Gamma_{j}}{\partial X_{k}}\Delta n_{j}}} \\ \vdots \\{\frac{\partial^{2}n_{1}}{{\partial\omega}{\partial X_{K}}} + {\frac{\partial}{\partial\omega}\Sigma_{j}\frac{\partial\Gamma_{j}}{\partial X_{k}}\Delta n_{j}}} \\ \vdots \\{\frac{\partial^{2}n_{1}}{\partial X_{K}^{2}} + {\Sigma_{j}\frac{\partial^{2}\Gamma_{j}}{\partial X_{k}^{2}}\Delta n_{j}}}\end{bmatrix}} & \left( {{Eq}.25} \right)\end{matrix}$

Photonic switches 2500 and 2600 illustrated in FIGS. 25 and 26 ,respectively, and the associated compensation equations are two examplesof how compensation structures can be implemented in myriad MZI-basedphotonic devices. One of skill in the art with the benefit of thisdisclosure can appreciate that similar compensation structures can beimplemented in other MZI-based photonic devices.

For simplicity, various components, such as the optical pump circuitry,substrates, cladding, and other components of MZI filter 100 (see FIG. 1) are not shown in the figures. In the foregoing specification,embodiments of the disclosure have been described with reference tonumerous specific details that can vary from implementation toimplementation. The specification and drawings are, accordingly, to beregarded in an illustrative rather than a restrictive sense. The soleand exclusive indicator of the scope of the disclosure, and what isintended by the applicants to be the scope of the disclosure, is theliteral and equivalent scope of the set of claims that issue from thisapplication, in the specific form in which such claims issue, includingany subsequent correction. The specific details of particularembodiments can be combined in any suitable manner without departingfrom the spirit and scope of embodiments of the disclosure.

Additionally, spatially relative terms, such as “bottom or “top” and thelike can be used to describe an element and/or feature's relationship toanother element(s) and/or feature(s) as, for example, illustrated in thefigures. It will be understood that the spatially relative terms areintended to encompass different orientations of the device in use and/oroperation in addition to the orientation depicted in the figures. Forexample, if the device in the figures is turned over, elements describedas a “bottom” surface can then be oriented “above” other elements orfeatures. The device can be otherwise oriented (e.g., rotated 90 degreesor at other orientations) and the spatially relative descriptors usedherein interpreted accordingly.

What is claimed is:
 1. An optical filter comprising: a first waveguideincluding a first region extending between a first coupler section and asecond coupler section, and a second region extending between the secondcoupler section and a third coupler section; and a second waveguideincluding: a first portion extending between the first coupler sectionand the second coupler section, the first portion including at least twocompensation sections that sequentially increase in width; and a secondportion extending between the second coupler section and the thirdcoupler section, the second portion including at least two compensationsections that sequentially increase in width.
 2. The optical filter ofclaim 1 wherein the first, second and third coupler sections and thefirst and second waveguides comprise a Mach-Zehnder interferometer (MZI)filter.
 3. The optical filter of claim 1 wherein the first portionincludes at least three compensation sections that sequentially increasein width.
 4. The optical filter of claim 1 wherein the second portionincludes at least three compensation sections that sequentially increasein width.
 5. The optical filter of claim 1 wherein the first portionincludes at least two compensation sections that sequentially decreasein width.
 6. The optical filter of claim 1 wherein the second portionincludes at least two compensation sections that sequentially decreasein width.
 7. The optical filter of claim 1 wherein the at least twocompensation sections of the first portion and the at least twocompensation sections of the second portion include respective taperportions positioned between each of the increases in width of the secondwaveguide.
 8. The optical filter of claim 1 wherein the at least twocompensation sections of the first portion and the at least twocompensation sections of the second portion are sized and arranged toreduce a shift in a frequency response of the optical filter due tovariations in a width of the first waveguide and variations in a widthof the second waveguide.
 9. The optical filter of claim 1 wherein eachof the first portion and the second portion have a number of waveguidewidths that are greater than a predetermined number of perturbations.10. A method comprising: fabricating a first waveguide including a firstregion extending between a first coupler section and a second couplersection, and a second region extending between the second couplersection and a third coupler section; and fabricating a second waveguideincluding: a first portion extending between the first coupler sectionand the second coupler section, the first portion including at least twocompensation sections that sequentially increase in width; and a secondportion extending between the second coupler section and the thirdcoupler section, the second portion including at least two compensationsections that sequentially increase in width.
 11. The method of claim 10wherein the first, second and third coupler sections and the first andsecond waveguides comprise a Mach-Zehnder interferometer (MZI) filter.12. The method of claim 10 wherein the first region and the secondregion each have constant widths.
 13. The method of claim 10 wherein thefirst portion includes at least three compensation sections thatsequentially increase in width.
 14. The method of claim 10 wherein thesecond portion includes at least three compensation sections thatsequentially increase in width.
 15. The method of claim 10 wherein thefirst portion includes at least two compensation sections thatsequentially decrease in width.
 16. The method of claim 10 wherein thesecond portion includes at least two compensation sections thatsequentially decrease in width.
 17. The method of claim 10 wherein theat least two compensation sections of the first portion and the at leasttwo compensation sections of the second portion include respective taperportions positioned between each of the increases in width of the secondwaveguide.
 18. The method of claim 10 wherein the at least twocompensation sections of the first portion and the at least twocompensation sections of the second portion are sized and arranged toreduce a shift in a frequency response of an optical filter due tovariations in a width of the first waveguide and variations in a widthof the second waveguide.
 19. The method of claim 10 wherein each of thefirst portion and the second portion have a number of waveguide widthsthat are greater than a predetermined number of perturbations.
 20. Themethod of claim 10 wherein a length of the second region is twice alength of the first region.